For an algebra $\mathcal A$ and a complete lattice $\mathcal L$, one can consider a fuzzy congruence relation $\bar{\rho}$ (defined in [1]). Here we define a quotient algebra $\mathcal A/\rho$. Since every fuzzy congruence relation is a special union of a family of ordinary congruences on the same algebra, it is interesting to consider the relationship between $\mathcal A/\bar{\rho}$ and the quotient algebra $\mathcal A/\rho$ by any of the congruences of the family. We prove that there is always a homomorphism from $\mathcal A/\bar{\rho}$ to $\mathcal A/\rho$, and we give the necessary and sufficient conditions for it to be an isomorphism. We also consider the fuzzy subalgebras (defined as in [2]) of $\mathcal A$, and $\mathcal A/\rho$, and assuming that these mappings preserve the homomorphism, we prove that a fuzzy subalgebra $\bar{\mathcal A}$ of $\mathcal A$ induces $\bar{\mathcal A}/\rho$ (of $\mathcal A/\rho$) and vice versa. Using the homomorphism from $\mathcal A/\bar{\rho}$ on to $\mathcal A/\rho$ , we finally determine the connection between the corresponding fuzzy subalgebras.